scholarly journals On a problem in the theory of ordered groups

1972 ◽  
Vol 6 (3) ◽  
pp. 435-438 ◽  
Author(s):  
Colin D. Fox
Keyword(s):  
Cambridge Philos ◽  
Orderable Group ◽  
Counter Example ◽  
Ordered Groups ◽  
Defining Relation

The group G presented on two generators a, c with the single defining relation a−1c2a = c2a2c2 [proposed by B.H. Neumann in 1949 (unpublished), discussed by Gilbert Baumslag in Proc. Cambridge Philos. Soc. 55 (1959)] has been considered as a possible example of an orderable group which can not be embedded in a divisible orderable group, contrary to the conjecture that no such examples exist. It is known from Baumslag's discussion that G can not be embedded in any divisible orderable group. However, it is shown in this note that G is not orderable, and thus is not a counter-example to the conjecture.

Non-amalgamation of ordered groups

1984 ◽  
Vol 95 (2) ◽  
pp. 191-195 ◽  
Author(s):  
A. M. W. Glass ◽  
D. Saracino ◽  
C. Wood
Keyword(s):  
Positive Integer ◽  
Total Order ◽  
Ordered Group ◽  
Orderable Group ◽  
Ordered Groups

An ordered group (o-group for short) is a group endowed with a linear (i.e. total) order such that for all x, y, z, xz ≤ yz and zx ≤ zy whenever x ≤ y. A group for which such an order exists is called an orderable group. A group G is said to be divisible if for each positive integer m and each g ε G, there is x ε G such that xm = g.

scholarly journals Ordered groups, eigenvalues, knots, surgery and L-spaces

2011 ◽  
Vol 152 (1) ◽  
pp. 115-129 ◽  
Author(s):  
ADAM CLAY ◽  
DALE ROLFSEN
Keyword(s):  
Knot Theory ◽  
Real Root ◽  
Floer Homology ◽  
Heegaard Floer Homology ◽  
Necessary Condition ◽  
Finitely Generated ◽  
Positive Real ◽  
Orderable Group ◽  
Positive Real Root ◽  
Ordered Groups

AbstractWe establish a necessary condition that an automorphism of a nontrivial finitely generated bi-orderable group can preserve a bi-ordering: at least one of its eigenvalues, suitably defined, must be real and positive. Applications are given to knot theory, spaces which fibre over the circle and to the Heegaard–Floer homology of surgery manifolds. In particular, we show that if a nontrivial fibred knot has bi-orderable knot group, then its Alexander polynomial has a positive real root. This implies that many specific knot groups are not bi-orderable. We also show that if the group of a nontrivial knot is bi-orderable, surgery on the knot cannot produce an L-space, as defined by Ozsváth and Szabó.

scholarly journals An embedding theorem for ordered groups

1975 ◽  
Vol 12 (3) ◽  
pp. 321-335 ◽  
Author(s):  
Colin D. Fox
Keyword(s):  
Positive Integer ◽  
Embedding Theorem ◽  
Normal Closure ◽  
Orderable Group ◽  
Ordered Groups

We show that if the normal, closure of an element a, of an orderable group, G, is abelian, then G can be embedded in an orderable group, G#, which contains an n-th root of a for every positive integer, n. Furthermore, every order of G extends to an order of G#.

scholarly journals Roberts’ weak welfarism theorem: a minor correction

2021 ◽  
Author(s):  
Peter J. Hammond
Keyword(s):  
Euclidean Space ◽  
Social Welfare ◽  
Utility Function ◽  
Weak Continuity ◽  
Strict Preference ◽  
Counter Example ◽  
Independence Of Irrelevant Alternatives ◽  
Unrestricted Domain ◽  
A Minor ◽  
Minor Correction

AbstractRoberts’ “weak neutrality” or “weak welfarism” theorem concerns Sen social welfare functionals which are defined on an unrestricted domain of utility function profiles and satisfy independence of irrelevant alternatives, the Pareto condition, and a form of weak continuity. Roberts (Rev Econ Stud 47(2):421–439, 1980) claimed that the induced welfare ordering on social states has a one-way representation by a continuous, monotonic real-valued welfare function defined on the Euclidean space of interpersonal utility vectors—that is, an increase in this welfare function is sufficient, but may not be necessary, for social strict preference. A counter-example shows that weak continuity is insufficient; a minor strengthening to pairwise continuity is proposed instead and its sufficiency demonstrated.

Generalized Commutativity of Lattice-Ordered Groups

2015 ◽  
Vol 65 (2) ◽  
Author(s):  
M. R. Darnel ◽  
W. C. Holland ◽  
H. Pajoohesh
Keyword(s):  
Theorem Proving ◽  
Natural Generalization ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Weak Commutativity

AbstractIn this paper we explore generalizations of Neumann’s theorem proving that weak commutativity in ordered groups actually implies the group is abelian. We show that a natural generalization of Neumann’s weak commutativity holds for certain Scrimger ℓ-groups.

scholarly journals The Annihilation of Space: A Bad (Historical) Concept

2021 ◽  
pp. 1-30
Author(s):  
Alexis D. Litvine
Keyword(s):  
Cultural History ◽  
Intellectual History ◽  
Social History ◽  
Economic Consequences ◽  
Cultural Mediation ◽  
Counter Example ◽  
Dead End ◽  
The Social ◽  
History Of ◽  
Representations Of Space

Abstract This article is a reminder that the concept of ‘annihilation of space’ or ‘spatial compression’, often used as a shorthand for referring to the cultural or economic consequences of industrial mobility, has a long intellectual history. The concept thus comes loaded with a specific outlook on the experience of modernity, which is – I argue – unsuitable for any cultural or social history of space. This article outlines the etymology of the concept and shows: first, that the historical phenomena it pretends to describe are too complex for such a simplistic signpost; and, second, that the term is never a neutral descriptor but always an engagement with a form of historical and cultural mediation on the nature of modernity in relation to space. In both cases this term obfuscates more than it reveals. As a counter-example, I look at the effect of the railways on popular representations of space and conclude that postmodern geography is a relative dead end for historians interested in the social and cultural history of space.

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